On properties of a semi-explicit in time fourth-order vector compact scheme for the multidimensional acoustic wave equation

We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove stability and derive 4th order error bound in the enlarged energy norm. This scheme is three-point in each spatial direction, and it exploits additional sought functions which approximate 2nd order non-mixed spatial derivatives of the solution to the equation. At the first time level, a similar two-level in time scheme is applied, with no derivatives of the data. No iterations are required to implement the scheme. We also present results of various 3D numerical experiments that demonstrate a very high accuracy of the scheme for smooth data, its advantages in the error behavior over the classical explicit 2nd order scheme for nonsmooth data as well and an example of the wave in a layered medium initiated by the Ricker-type wavelet source function.